Gain of analyticity for semilinear Schroedinger equations

TL;DR

This paper proves that solutions to certain semilinear Schrödinger equations become real-analytic in space and Gevrey class in time if the initial data decays exponentially, revealing a gain of analyticity phenomenon.

Contribution

It establishes the gain of analyticity for solutions with exponentially decaying initial data, extending previous energy estimate methods to this context.

Findings

Solutions become real-analytic in space

Solutions are Gevrey class in time

Analyticity gain occurs except at initial plane

Abstract

We discuss gain of analyticity phenomenon of solutions to the initial value problem for semilinear Schroedinger equations with gauge invariant nonlinearity. We prove that if the initial data decays exponentially, then the solution becomes real-analytic in the space variable and a Gevrey function of order 2 in the time variable except in the initial plane. Our proof is based on the energy estimates developed in our previous work and on fine summation formulae concerned with a matrix norm.